Distinguished tame supercuspidal representations and odd orthogonal periods

Hakim, Jeffrey; Lansky, Joshua

doi:10.1090/S1088-4165-2012-00418-6

Distinguished tame supercuspidal representations and odd orthogonal periods
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by Jeffrey Hakim and Joshua Lansky

Represent. Theory 16 (2012), 276-316

DOI: https://doi.org/10.1090/S1088-4165-2012-00418-6

Published electronically: June 1, 2012

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Abstract:

We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of $\mathrm {GL}_n(F)$, with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distinguishing linear forms has dimension one.

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